Original Paper

The stationary set splitting game

  1. Paul B. Larson,
  2. Saharon Shelah2,3

Article first published online: 7 MAR 2008

DOI: 10.1002/malq.200610054

Issue

Mathematical Logic Quarterly

Mathematical Logic Quarterly

Volume 54, Issue 2, pages 187–193, April 2008

Additional Information

How to Cite

Larson, P. B. and Shelah, S. (2008), The stationary set splitting game. Mathematical Logic Quarterly, 54: 187–193. doi: 10.1002/malq.200610054

Author Information

  1. 2

    The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, Jersalem 91904, Israel

  2. 3

    Department of Mathematics, Hill Center – Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA

Publication History

  1. Issue published online: 7 MAR 2008
  2. Article first published online: 7 MAR 2008
  3. Manuscript Accepted: 29 AUG 2007
  4. Manuscript Revised: 17 AUG 2007
  5. Manuscript Received: 6 DEC 2006

Funded by

  • NSF. Grant Numbers: DMS-0401603, DMS-0600940
  • United States-Israel Binational Science Foundation

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Keywords:

  • Definable determinacy;
  • games of uncountable length

Abstract

The stationary set splitting game is a game of perfect information of length ω1 between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ22 maximality with a predicate for the nonstationary ideal on ω1, and an example of a consistently undetermined game of length ω1 with payoff de.nable in the second-order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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